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Mirrors > Home > MPE Home > Th. List > sbalv | Structured version Visualization version GIF version |
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sbalv.1 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbalv | ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbal 2159 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑) | |
2 | sbalv.1 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | |
3 | 2 | albii 1822 | . 2 ⊢ (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓) |
4 | 1, 3 | bitri 274 | 1 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1537 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-11 2154 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-sb 2068 |
This theorem is referenced by: sbex 2278 sbmo 2616 sbabel 2941 sbabelOLD 2942 mo5f 30823 |
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